Lesson 7: The Derivative (handout)

Many functions in nature are described as the rate of change of another function. The concept is called the derivative. Algebraically, the process of finding the derivative involves a limit of difference quotients.

Lesson 7: The Derivative (handout)

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. V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Notes Sec on 2.1–2.2 The Deriva ve V63.0121.001: Calculus I Professor Ma hew Leingang New York University . February 14, 2011 . NYUMathematics . Notes Announcements Quiz this week on Sec ons 1.1–1.4 No class Monday, February 21 . . Objectives Notes The Derivative Understand and state the deﬁni on of the deriva ve of a func on at a point. Given a func on and a point in its domain, decide if the func on is diﬀeren able at the point and ﬁnd the value of the deriva ve at that point. Understand and give several examples of deriva ves modeling rates of change in science. . . . 1.

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. V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Objectives Notes The Derivative as a Function Given a func on f, use the deﬁni on of the deriva ve to ﬁnd the deriva ve func on f’. Given a func on, ﬁnd its second deriva ve. Given the graph of a func on, sketch the graph of its deriva ve. . . Notes Outline Rates of Change Tangent Lines Velocity Popula on growth Marginal costs The deriva ve, deﬁned Deriva ves of (some) power func ons What does f tell you about f′ ? How can a func on fail to be diﬀeren able? Other nota ons The second deriva ve . . The tangent problem Notes A geometric rate of change Problem Given a curve and a point on the curve, ﬁnd the slope of the line tangent to the curve at that point. Solu on . . . 2.

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. V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Population growth Notes Biological Rates of Change Problem Given the popula on func on of a group of organisms, ﬁnd the rate of growth of the popula on at a par cular instant. Solu on . . Notes Population growth example Example Suppose the popula on of ﬁsh in the East River is given by the func on 3et P(t) = 1 + et where t is in years since 2000 and P is in millions of ﬁsh. Is the ﬁsh popula on growing fastest in 1990, 2000, or 2010? (Es mate numerically) Answer . . Notes Derivation Solu on Let ∆t be an increment in me and ∆P the corresponding change in popula on: ∆P = P(t + ∆t) − P(t) This depends on ∆t, so ideally we would want ( ) ∆P 1 3et+∆t 3et lim = lim − ∆t→0 ∆t ∆t→0 ∆t 1 + et+∆t 1 + et But rather than compute a complicated limit analy cally, let us approximate numerically. We will try a small ∆t, for instance 0.1. . . . 5.

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. V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Notes What does f tell you about f′? If f is a func on, we can compute the deriva ve f′ (x) at each point x where f is diﬀeren able, and come up with another func on, the deriva ve func on. What can we say about this func on f′ ? If f is decreasing on an interval, f′ is nega ve (technically, nonposi ve) on that interval If f is increasing on an interval, f′ is posi ve (technically, nonnega ve) on that interval . . Notes What does f tell you about f′? Fact If f is decreasing on the open interval (a, b), then f′ ≤ 0 on (a, b). Picture Proof. If f is decreasing, then all secant lines point downward, hence have y nega ve slope. The deriva ve is a limit of slopes of secant lines, which are all nega ve, so the limit must be ≤ 0. . x . . Notes What does f tell you about f′? Fact If f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b). The Real Proof. If ∆x > 0, then f(x + ∆x) − f(x) f(x + ∆x) < f(x) =⇒ <0 ∆x If ∆x < 0, then x + ∆x < x, and f(x + ∆x) − f(x) f(x + ∆x) > f(x) =⇒ <0 . ∆x . . 9.

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. V63.0121.001: Calculus I . Sec on 2.1–2.2: The Deriva ve . February 14, 2011 Notes What does f tell you about f′? Fact If f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b). The Real Proof. f(x + ∆x) − f(x) Either way, < 0, so ∆x f(x + ∆x) − f(x) f′ (x) = lim ≤0 ∆x→0 ∆x . . Notes Going the Other Way? Ques on If a func on has a nega ve deriva ve on an interval, must it be decreasing on that interval? Answer Maybe. . . Notes Outline Rates of Change Tangent Lines Velocity Popula on growth Marginal costs The deriva ve, deﬁned Deriva ves of (some) power func ons What does f tell you about f′ ? How can a func on fail to be diﬀeren able? Other nota ons The second deriva ve . . . 10.